Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $t = \dfrac{-3p + 15}{5p - 45} \times \dfrac{p^2 + 10p + 21}{p^2 - 2p - 15} $
First factor out any common factors. $t = \dfrac{-3(p - 5)}{5(p - 9)} \times \dfrac{p^2 + 10p + 21}{p^2 - 2p - 15} $ Then factor the quadratic expressions. $t = \dfrac {-3(p - 5)} {5(p - 9)} \times \dfrac {(p + 3)(p + 7)} {(p + 3)(p - 5)} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac {-3(p - 5) \times (p + 3)(p + 7) } {5(p - 9) \times (p + 3)(p - 5) } $ $t = \dfrac {-3(p + 3)(p + 7)(p - 5)} {5(p + 3)(p - 5)(p - 9)} $ Notice that $(p + 3)$ and $(p - 5)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac {-3\cancel{(p + 3)}(p + 7)(p - 5)} {5\cancel{(p + 3)}(p - 5)(p - 9)} $ We are dividing by $p + 3$ , so $p + 3 \neq 0$ Therefore, $p \neq -3$ $t = \dfrac {-3\cancel{(p + 3)}(p + 7)\cancel{(p - 5)}} {5\cancel{(p + 3)}\cancel{(p - 5)}(p - 9)} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $t = \dfrac {-3(p + 7)} {5(p - 9)} $ $ t = \dfrac{-3(p + 7)}{5(p - 9)}; p \neq -3; p \neq 5 $